Question

Frank is having a raised rectangular platform built. He designed it so that the length of the platform is 4 feet longer than the width, w, and the height, h, of the platform is 10 feet less than the width of the platform. The cost of the raised platform will be based on its top area and its height. The cost of the raised platform is $8 per square foot of the top area plus $4 per foot of the height, for a total cost of $1,544. Create a system of equations to model the situation above. Determine the solutions and if the solutions are viable. If there is one viable solution, fill in the values below for the width, length, and height of the raised platform. If there is more than one viable solution, choose the solution with the smallest width, and fill in the values below for the width, length, and height of the raised platform. If there is no viable solution, fill in each blank with the word "no".

Answer 1

One system of equations would be
L = w+4
H = w-10
4H + 8(wL) = 1544
There is one viable solution; the width is 12, the length is 16, and the height is 2.

Using substitution with the system of equations, we have
4(w-10)+8(w(w+4))=1544
4w-40+8(w²+4w)=1544
4w-40+8w²+32w=1544

Combining like terms, we have
8w²+36w-40=1544

Factoring out a 4, we have
4(2w²+9w-10)=1544

Dividing both sides by 4 gives us
2w²+9w-10=386

Subtract 386 from both sides to get
2w²+9w-396=0

Using the quadratic formula, we have
$w=\frac{-9\pm \sqrt{9^2-4(2)(-396)}}{2(2)} \\ \\=\frac{-9\pm \sqrt{81--3168}}{4} \\ \\=\frac{-9\pm \sqrt{3249}}{4}=\frac{-9\pm 57}{4} \\ \\=\frac{-9+57}{4}\text{ or }\frac{-9-57}{4}=\frac{48}{4}\text{ or }\frac{-66}{4} \\ \\=12\text{ or }-16.5$

Since a negative width makes no sense, we know that w=12.

This means L=w+4=12+4=16 and H=w-10=12-10=2